(ed.). Gravitational waves (IOP, 2001)(422s).

Cosmological perturbation theory 303we get, for each mode (now generically denoted with h), the effective scalar actionδ (2) S = 1 ∫ (d 4 xa 3 e −φ ḣ 2 + h ∇ )2a 2 h , (16.71)which can be rewritten, using conformal time, as∫δ (2) S = 1 2d 3 x dη a 2 e −φ (h ′2 + h∇h). (16.72)The variation with respect to h gives finally equation (16.54), i.e. the sameequation obtained by perturbing directly the background equations in the S-frame.The above action describes a scalar field h, non-minimally coupled to a timedependentexternal field, a 2 e −φ (also called ‘pump field’). In order to impose thecorrect quantum normalization to vacuum fluctuations, we introduce now the socalled‘canonical variable’ ψ, defined in terms of the pump field asψ = zh, z = ae −φ/2 . (16.73)With such a definition the kinetic term for ψ appears in the standard canonicalform: for each mode k, in fact, we get the actionδ (2) S k = 1 ∫ ()dη ψ k′22− k2 ψk 2 + z′′z ψ2 k , (16.74)and the corresponding canonical evolution equation:ψ ′′k + [k2 − V (η)]ψ k = 0,V (η) = z′′z , (16.75)which has the form of a Schrodinger-like equation, with an effective potentialdepending on the external pump field. This form of the canonical equation, bythe way, is the same for all types of perturbations (with different potentials, ofcourse). What is important, in our context, is that for an accelerated inflationarybackground V (z) → 0asη → −∞. This means that, asymptotically, thecanonical variable satisfies the free-field oscillating equationη →−∞, ψ ′′k + k2 ψ k = 0, (16.76)and can be normalized to an initial vacuum fluctuation spectrum,η →−∞, ψ k = 1 √2ke −ikη , (16.77)in such a way as to satisfy the free field canonical commutation relations,[ψ k ,ψ ∗′j] = iδ kj . The normalization of ψ k then fixes the normalization of themetric variable h k = ψ k /z.